Prove of the complicated integral

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The discussion focuses on proving the integral inequality \(\frac{2}{e^{5}} \leq \int\int_{D} e^{-(x^{2}+y^{2})} dx dy \leq 2\) over the domain \(D = [0,1] \times [0,2]\). Participants suggest using the change of variables theorem due to the diffeomorphic relationship between the rectangle and a circle. The maximum value of the function \(e^{-(x^2+y^2)}\) is established as 1, occurring at the origin (0,0). The integral's bounds are confirmed through analysis of the function's behavior within the specified domain.

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Hi experts!

How to prove this integral?

[itex]\frac{2}{e^{5}}[/itex][itex]\leq[/itex][itex]\int[/itex][itex]\int_{D}[/itex]e[itex]^{-(x^{2}+y^{2})}[/itex]dxdy[itex]\leq[/itex]2
on
D=[0,1] and [0,2]

Here D is subscript of the 2nd integral.

I seriously have no idea how to start. I am 100% blank with this question.

Thanks in advance.
 
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What is the maximum/minimum value that [itex]e^{-(x^2+y^2)}[/itex] can attain??
 
Why not just calculate the integral? The rectangle (0,1) x (0,2) is diffeomorphic to a circle (or the circle minus a set of measure zero) so use the change of variables theorem.
 

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